Time inhomogeneous quantum dynamical maps

We discuss a wide class of time inhomogeneous quantum evolution which is represented by two-parameter family of completely positive trace-preserving maps. These dynamical maps are constructed as infinite series of jump processes. It is shown that such dynamical maps satisfy time inhomogeneous memory kernel master equation which provides a generalization of the master equation involving the standard convolution. Time-local (time convolution-less) approach is discussed as well. Finally, the comparative analysis of traditional time homogeneous versus time inhomogeneous scenario is provided.


Time inhomogeneous quantum dynamical maps Dariusz Chruściński
We discuss a wide class of time inhomogeneous quantum evolution which is represented by two-parameter family of completely positive trace-preserving maps. These dynamical maps are constructed as infinite series of jump processes. It is shown that such dynamical maps satisfy time inhomogeneous memory kernel master equation which provides a generalization of the master equation involving the standard convolution. Time-local (time convolution-less) approach is discussed as well. Finally, the comparative analysis of traditional time homogeneous versus time inhomogeneous scenario is provided.
The dynamics of an open quantum system 1,2 is usually represented by the dynamical map { t,t 0 } t≥t 0 , i.e. a family of completely positive trace-preserving maps � t,t 0 : B(H) → B(H) 3,4 ( B(H) stands for the vector space of bounded linear operators acting on the system's Hilbert space H ). In this paper we consider only finite dimensional scenario and hence B(H) contains all linear operators. The map t,t 0 transforms any initial system's state represented by a density operator ρ 0 at an initial time t 0 into a state at the current time t, i.e. ρ t = � t,t 0 (ρ 0 ) . Dynamical maps { t,t 0 } t≥t 0 provide the powerful generalization of the standard Schrödinger unitary evolution U t,t 0 ρ 0 U † t,t 0 , where U t,t 0 is a family of unitary operators acting on H . A dynamical map is usually realized as a reduced evolution 1 where U t,t 0 is a unitary operator acting on H ⊗ H E , ρ E is a fixed state of the environment (living in H E ), and Tr E denotes a partial trace (over the environmental degrees of freedom). The unitary U t,t 0 is governed by the total (in general time-dependent) 'system + environment' Hamiltonian H t . Now, if H t = H does not depend on time the reduced evolution (1.1) is time homogeneous (or translationally invariant), i.e. t,t 0 = t−t 0 (or equivalently � t+τ ,t 0 +τ = � t,t 0 for any τ ). In this case one usually fixes t 0 = 0 and simply considers one-parameter family of maps { t } t≥0 . Such scenario is usually considered by majority of authors. The most prominent example of time homogeneous dynamical maps is the celebrated Markovian semigroup t = e Lt , where L denotes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) generator 5,6 (cf. also the detailed exposition in 7 and 8 for a brief history) with the (effective) system's Hamiltonian H, noise operators L k , and non-negative transition rates γ k . It is well known, however, that semigroup evolution usually requires a series of additional assumptions and approximations like e.g. weak system-environment interaction and separation of natural time scales of the system and environment. Departure from a semigroup scenario calls for more refined approach which attracts a lot of attention in recent years and is intimately connected with quantum non-Markovian memory effects (cf. recent reviews [9][10][11][12][13][14][15][16][17] ). To go beyond dynamical semigroup keeping translational invariance one replaces time independent GKLS generator L by a memory kernel {K t } t≥0 and considers the following dynamical equation where A • B denotes composition of two maps. Equation (1.3) is often referred as Nakajima-Zwanzig master equation 18,19 . The very structure of the convolution K t * t does guarantee translational invariance. However, the property of complete positivity of t is notoriously difficult as already observed in [20][21][22] . Time non-local master equation (1.3) were intensively studied by several authors [23][24][25][26][27][28][29][30][31][32][33][34][35] . Since the master equation (1.3) involving www.nature.com/scientificreports/ the convolution is technically quite involved one usually tries to describe the dynamics in terms of convolutionless time-local approach involving a time dependent generator {L t } t≥0 (cf. the recent comparative analysis 36 ). Time-local generator L t plays a key role in characterizing the property of CP-divisibility which is essential in the analysis of Markovianity. Note, however, that the corresponding propagator t,s = t • −1 s is no longer time homogeneous unless L t is time independent.

OPEN
In this paper we go beyond time homogeneous case and consider the following generalization of (1.3) which reduces to (1.3) if K t,τ = K t−τ . Equation (1.4) may be, therefore, considered as a time inhomogeneous Nakajima-Zwanzig master equation. Such description is essential whenever the 'system + environment' Hamiltonian H t does depend on time. Note, that formally if K t,τ = L t δ(t − τ ) , then (1.4) reduces to time-local but inhomogeneous master equation and the corresponding solution t,t 0 is CPTP for all t and t 0 with t > t 0 if and only if L t is of GKLS form for all t ∈ R 1,2,7 . This is just inhomogeneous generalization of semigroup evolution and it is often called an inhomogeneous semigroup 7 . Note, that contrary to the homogeneous scenario where the time-local generator is defined now for all t ∈ R. In this paper we propose a particular representation of dynamical maps { t,t 0 } t≥t 0 which by construction satisfy (1.4). Hence, it may be also considered as a particular construction of a legitimate class of memory kernels K t,τ giving rise to CPTP dynamical maps. Clearly, it is not the most general construction. However, the proposed representation possesses a natural physical interpretation in terms of quantum jumps. Time-local (time convolution-less) approach is discussed as well. It turns out that a time dependent generator also depends upon the initial time t 0 , i.e. one has a two-parameter family of generators {L t,t 0 } t≥t 0 . Finally, the comparative analysis of traditional time homogeneous versus time inhomogeneous scenario is provided.

Time homogeneous evolution
Markovian semigroup. Consider a Markovian semigroup governed by the time independent master equation where L stands for the GKLS generator (1.2), and t 0 is an arbitrary initial time. It is clear that since L does not depend on time the dynamical map depends upon the difference t − t 0 , i.e. the solution of (2.1) defines oneparameter semigroup � t,t 0 = � t−t 0 = e (t−t 0 )L . Usually, one assumes t 0 = 0 and simply writes t . Observe, that any GKLS generator (1. Proof let us introduce a perturbation parameter and a one-parameter family of generators such that L = L ( =1) . We find a solution to as a perturbation series Inserting the series (2.7) into (2.6) one finds the following infinite hierarchy of equations

4) is an alternative representation for the conventional exponential representation
Note, that contrary to (2.13) each term in (2.4) is completely positive and has a clear physical interpretation: an ℓ th term reads and it can be interpreted as follows: there are ℓ quantum jumps up to time 't' at {t 1 ≤ t 2 ≤ . . . ≤ t ℓ } represented by a completely positive map . Between jumps the system evolves according to (unperturbed) completely positive maps � (0) The series (2.4) represents all possible scenario of ℓ jumps for ℓ = 0, 1, 2, . . . . By construction, the resulting completely positive map t is also trace-preserving. One often calls (2.4) a quantum jump representation of a dynamical map [37][38][39] . Note, however, that truncating (2.4) at any finite ℓ violates tracepreservation since processes with more than ℓ jumps are not included. The standard exponential representation (2.13) does not have any clear interpretation. Each separate term t k L k does annihilate the trace but is not completely positive. Only the infinite sum of such terms gives rise to completely positive (and trace-preserving) map.
. . . (2.14) t defines a semigroup if and only if Z t = δ(t)Z . Consider a family of jump operators represented by completely positive maps { t } t≥0 . Define now the following generalization of (2.4) that is, one replaces � • �

Remark 1
Usually on solves the time homogeneous differential equations using the technique of Laplace transform. We provide the alternative proof of Proposition 2 in the Supplementary Information. Here, we provided the proof which can be easily generalized to inhomogeneous case where the Laplace transform technique can not be directly applied.

Remark 2
It is clear that if � (0) t = e −Zt is a semigroup, i.e. Z t = δ(t)Z , then � t = δ(t)� , and hence . . . www.nature.com/scientificreports/ Corollary 2 Introducing two completely positive maps Q t := � t * � (0) t and P t := � (0) t * � t a series (2.17) can be rewritten as follows or, equivalently, that is, one has exactly the same representation as in the case of semigroup (2.15). The only difference is the definition of Q t and P t in terms of t and � (0) i.e. one recovers the same relation as in Corollary 1.

Remark 3
It should be stressed that even when t is not completely positive, but Q t = � t * � with To find the corresponding jump representation of t,t 0 let us introduce the following (inhomogeneous) generalization of the convolution.

Definition 1 For any two families of maps A t,t 0 and B t,t 0
Note, that when A t,t 0 = A t−t 0 and B t,t 0 = B t−t 0 , then Proposition 3 The convolution (3.7) is associative for any three families A t,t 0 , B t,t 0 and C t,t 0 .
(2.29) � t = � (0) t + P t + P t * P t + P t * P t * P t + · · · * � (0) t , Proof the proof is a generalization of the proof of Proposition 1. Consider the family of generators We find a solution to as a perturbation series Inserting the series (3.13) into (3.12) one finds the following hierarchy of dynamical equations: with initial conditions Clearly, the above hierarchy provides a generalization of (2.8) for the inhomogeneous scenario. Now, defines an inhomogeneous semigroup which is completely positive (but not trace-preserving). As before it is sufficient to show that solves (3.14). One finds t,t 0 , one gets and finally, observing that one completes the proof.
For an alternative proof which does not use properties of the convolution ' ⊛ ' cf. Supplementary Information. Beyond an inhomogeneous semigroup. Suppose now that for any initial time � (0) t,t 0 is an arbitrary completely positive and trace non-increasing map satisfying � (0) t 0 ,t 0 = id . Let {Z t,t 0 } t≥t 0 be a family of maps such that that is {Z t,t 0 } t≥t 0 is a inhomogeneous generalization of {Z t } t≥0 . Now, Z t,t 0 does not only depends upon the current time 't' but also upon the initial time t 0 . Define the following generalization of (3.10) (3.14) with initial conditions (3.15). Clearly, the above hierarchy provides a generalization of (2.24) for the inhomogeneous scenario. It is enough to prove that One has Using � (0) t,t 0 , one gets and hence which ends the proof.

Time local approach
Very often describing the evolution of an open system one prefers to use a time-local (or so-called convolutionless (TCL)) approach 1 . Formally, in the time homogeneous case given a dynamical map { t } t≥0 one defines the corresponding time-local generator L t := [∂ t � t ] • � −1 t (assuming that t is invertible). This way the map t satisfies This procedure might be a bit confusing since (4.1) coincides with (3.1) for the inhomogeneous map t,t 0 . To clarify this point let us introduce again an initial time and consider t,t 0 = t−t 0 . Now, the time-local generator reads that is, the generator does depend upon the initial time 40 . It implies that the corresponding propagators also does depend upon t 0 . Clearly, fixing t 0 = 0 this fact is completely hidden. The dependence upon t 0 drops out only in the semigroup case when L t−t 0 = L. Similar analysis may be applied to inhomogeneous scenario as well. Now, instead of convolution (3.21) one may define a time-local generator such that t,t 0 satisfies the following inhomogeneous TCL master equation Again, the corresponding propagator also does depend upon t 0 . Hence, the local composition law holds only if the above propagators are defined w.r.t. the same initial time. Otherwise, composing the propagators does not have any sense. Equation (4.5) reduces to (3.1) only if L t,t 0 does not depend upon t 0 . In this case one recovers an inhomogeneous semigroup and L t,t 0 = L t .
Interestingly, apart from Nakajima-Zwanzing memory kernel master equation the map t,t 0 satisfies the following dynamical equation where the new kernel K t,t 0 is defined by that is, it is constructed in terms of the 'free' evolution represented by � (0) t,t 0 and the jump operators t,t 0 (the details of the derivation are presented in the Supplementary Information). This is very general class of legitimate quantum evolutions and corresponding dynamical equations. It would be interesting to apply the above scheme to discuss time inhomogeneous semi-Markov processes 28,29,33,41 and collision models (cf. 42 for the recent review).

Data availibility
All data generated or analysed during this study are included in this published article and its supplementary information file. t−t 0 + � (1) t−t 0 + � (2) t−t 0 + · · · ,